Cumulative trapezoidal numerical integration
Y is a vector, then
is the cumulative integral of
Y is a matrix, then
is the cumulative integral over each column.
Y is a multidimensional array, then
cumtrapz(Y) integrates over the first dimension whose
size does not equal 1.
X is a vector of coordinates, then
length(X) must be equal to the size of the first
Y whose size does not equal 1.
X is a scalar spacing, then
cumtrapz(X,Y) is equivalent to
Q = cumtrapz(___,
integrates along the dimension
dim using any of the previous
syntaxes. You must specify
Y, and optionally can specify
X. If you specify
X, then it can be a
scalar or a vector with length equal to
size(Y,dim). For example,
Y is a matrix, then
cumulatively integrates each row of
Calculate the cumulative integral of a vector where the spacing between data points is 1.
Create a numeric vector of data.
Y = [1 4 9 16 25];
Y contains function values for in the domain
Use cumtrapz to integrate the data with unit spacing.
Q = cumtrapz(Y)
Q = 1×5 0 2.5000 9.0000 21.5000 42.0000
This approximate integration yields a final value of 42. In this case, the exact answer is a little less, . The
cumtrapz function overestimates the value of the integral because f(x) is concave up.
Calculate the cumulative integral of a vector where the spacing between data points is uniform, but not equal to 1.
Create a domain vector.
X = 0:pi/5:pi;
Calculate the sine of
Y = sin(X');
cumtrapz. When the spacing between points is constant, but not equal to 1, an alternative to creating a vector for
X is to specify the scalar spacing value. In that case,
cumtrapz(pi/5,Y) is the same as
Q = cumtrapz(X,Y)
Q = 6×1 0 0.1847 0.6681 1.2657 1.7491 1.9338
Cumulatively integrate the rows of a matrix where the data has a nonuniform spacing.
Create a vector of x-coordinates and a matrix of observations that take place at the irregular intervals. The rows of
Y represent velocity data, taken at the times contained in
X, for three different trials.
X = [1 2.5 7 10]; Y = [5.2 7.7 9.6 13.2; 4.8 7.0 10.5 14.5; 4.9 6.5 10.2 13.8];
cumtrapz to integrate each row independently and find the cumulative distance traveled in each trial. Since the data is not evaluated at constant intervals, specify
X to indicate the spacing between the data points. Specify
dim = 2 since the data is in the rows of
Q1 = cumtrapz(X,Y,2)
Q1 = 3×4 0 9.6750 48.6000 82.8000 0 8.8500 48.2250 85.7250 0 8.5500 46.1250 82.1250
The result is a matrix of the same size as
Y with the cumulative integral of each row.
Perform nested integrations in the x and y directions. Plot the results to visualize the cumulative integral value in both directions.
Create a grid of values for the domain.
x = -2:0.1:2; y = -2:0.2:2; [X,Y] = meshgrid(x,y);
Calculate the function on the grid.
F = 10*X.^2 + 20*Y.^2;
cumtrapz integrates numeric data rather than functional expressions, so in general the underlying function does not need to be known to use
cumtrapz on a matrix of data. In cases where the functional expression is known, you can instead use
cumtrapz to approximate the double integral
To perform this double integration, use nested function calls to
cumtrapz. The inner call first integrates the rows of data, then the outer call integrates the columns.
I = cumtrapz(y,cumtrapz(x,F,2));
Plot the surface representing the original function as well as the surface representing the cumulative integration. Each point on the surface of the cumulative integration gives an intermediate value of the double integral. The last value in
I gives the overall approximation of the double integral,
I(end) = 642.4. Mark this point in the plot with a red star.
surf(X,Y,F,'EdgeColor','none') xlabel('X') ylabel('Y') hold on surf(X,Y,I,'FaceAlpha',0.5,'EdgeColor','none') plot3(X(end),Y(end),I(end),'r*') hold off
Y— Numeric data
Numeric data, specified as a vector, matrix, or multidimensional array. By
cumtrapz integrates along the first dimension of
Y whose size does not equal 1.
Complex Number Support: Yes
X— Point spacing
1(default) | uniform scalar spacing | vector of coordinates
Point spacing, specified as
1 (default), a uniform
scalar spacing, or a vector of coordinates.
X is a scalar, then it specifies a uniform
spacing between the data points and
is equivalent to
X is a vector, then it specifies
x-coordinates for the data points and
length(X) must be the same as the size of the
integration dimension in
dim— Dimension to operate along
Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.
Consider a two-dimensional input array,
cumtrapz(Y,1) works on successive elements
in the columns of
cumtrapz(Y,2) works on successive elements
in the rows of
dim is greater than
cumtrapz returns an array of zeros of the same size
cumtrapz to perform
numerical integrations on discrete data sets. Use
integral3 instead if
a functional expression for the data is available.
trapz reduces the size of the dimension it operates on to
1, and returns only the final integration value.
also returns the intermediate integration values, preserving the size of the
dimension it operates on.